Towards an Efficient Bisection of Ellipsoids

  • Paden Portillo
  • Martine Ceberio
  • Vladik Kreinovich
Part of the Studies in Computational Intelligence book series (SCI, volume 539)


Constraints are often represented as ellipsoids. One of the main advantages of such constrains is that, in contrast to boxes, over which optimization of even quadratic functions is NP-hard, optimization of a quadratic function over an ellipsoid is feasible. Sometimes, the area described by constrains is too large, so it is reasonable to bisect this area (one or several times) and solve the optimization problem for all the sub-areas. Bisecting a box, we still get a box, but bisecting an ellipsoid, we do not get an ellipsoid. Usually, this problem is solved by enclosing the half-ellipsoid in a larger ellipsoid, but this slows down the domain reduction process. Instead, we propose to optimize the objective functions over the resulting half-, quarter, etc., ellipsoids.


constraints ellipsoids bisection computational complexity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Belforte, G., Bona, B.: An improved parameter identification algorithm for signal with unknown-but-bounded errors. In: Proceedings of the 7th IFAC Symposium on Identification and Parameter Estimation, NewYork, U.K. (1985)Google Scholar
  2. 2.
    Chernousko, F.L.: Estimation of the Phase Space of Dynamic Systems. Nauka Publ., Moscow (1988) (in Russian)Google Scholar
  3. 3.
    Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)Google Scholar
  4. 4.
    Cormen, C.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Boston (2009)MATHGoogle Scholar
  5. 5.
    Ferson, S., Ginzburg, L., Kreinovich, V., Longpré, L., Aviles, M.: Exact bounds on finite populations of interval data. Reliable Computing 11(3), 207–233 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Filippov, A.F.: Ellipsoidal estimates for a solution of a system of differential equations. Interval Computations 2(2(4)), 6–17 (1992)Google Scholar
  7. 7.
    Fogel, E., Huang, Y.F.: On the value of information in system identification. Bounded noise case. Automatica 18(2), 229–238 (1982)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Interval computations website,
  9. 9.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001)MATHGoogle Scholar
  10. 10.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–396 (1984)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht (1998)CrossRefMATHGoogle Scholar
  12. 12.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM Press, Philadelphia (2009)CrossRefMATHGoogle Scholar
  13. 13.
    Schweppe, F.C.: Recursive state estimation: unknown but bounded errors and system inputs. IEEE Transactions on Automatic Control 13, 22 (1968)CrossRefGoogle Scholar
  14. 14.
    Schweppe, F.C.: Uncertain Dynamic Systems. Prentice Hall, Englewood Cliffs (1973)Google Scholar
  15. 15.
    Vavasis, S.A.: Nonlinear Optimization: Complexity Issues. Oxford University Press, New York (1991)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Paden Portillo
    • 1
  • Martine Ceberio
    • 1
  • Vladik Kreinovich
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas at El PasoEl PasoUSA

Personalised recommendations